Domain Wall Six-Vertex Model with Half-Turn Symmetry

Abstract

We obtain asymptotic formulas for the partition function of the six-vertex model with domain wall boundary conditions and half-turn symmetry in each of the phase regions. The proof is based on the Izergin-Korepin-Kuperberg determinantal formula for the partition function, its reduction to orthogonal polynomials, and on an asymptotic analysis of the orthogonal polynomials under consideration in the framework of the Riemann-Hilbert approach.

Phase separation in the six-vertex model with a variety of boundary conditions

“…• Arctic curves in the six-vertex model with HTBC and η = 0 (DWBC with half turn symmetry [29]) are found to be, in the thermodynamic limit, the lower half of the known arctic curves [18,19] with DWBC, in all the cases we examined (∆ = 0, ∆ = −2, b = a, b = 2a). The same is true at ∆ = 0 for the vertical As a future perspective, it would be natural to investigate what is the more general class of boundary conditions that lead to the same arctic curves in the thermodynamic limit as DWBC.…”

“…9b. In hindsight, this is perhaps not surprising, in [29] for instance it is proved that the extensive part of the the free-energy is the same for the two models in the thermodynamic limit. Finally, let us now denote by h s (x) the vertical edge density computed on the straight line r s of Fig.…”

“…We modified the original AR algorithm, to address the study of pDWBC, the six-vertex model with a RE and HTBC. In some cases [23,28,29], exact results for the partition functions were known in the thermodynamic limit but the presence of phase separation and the shape of the arctic curve have been never investigated. We show that when the model has only two parameters a and b, typical configurations display features similar to DWBC, varying ∆ in Eq.…”

“…The interface becomes less fluctuating and more straight when ∆ decreases. [29] are also simply related if one assumes w(a 1 ) = w(a 2 ) = a, w(b 1 ) = w(b 2 ) = b, w(c 1 ) = w(c 2 ) = 1 as we will also do in the following. However, notice that as for pDWBC the six-vertex model with HTBC would have in principle an extra parameter defined from w(a 1 )/w(a 2 ) = w(b 1 )/w(b 2 ) = e −2η .…”

“…For instance, analogous boundary conditions, especially related to alternating-sign matrix enumeration problems, have appeared in the literature [25][26][27] for which partition functions could be expressed through suitable determinants or pfaffians. Moreover the thermodynamic limit of the partition function has been calculated for half-turn boundary conditions when w(a 1 ) = w(a 2 ) and w(b 1 ) = w(b 2 ) (or DWBC with half turn symmetry, see [29]) and partial domain wall boundary conditions in ferromagnetic phase [28]. In the former case the extensive part of the free energy is the same as for DWBC.…”